A normal version of Brauer's height zero conjecture

TL;DR

This paper extends the understanding of Brauer's height zero conjecture by isolating the normal part in the context of two primes, providing new insights and an alternative proof for the conjecture.

Contribution

It demonstrates that the normal part of the conjecture can be isolated similarly to the abelian Sylow subgroup case, based on work by Malle and Navarro.

Findings

Isolation of the normal part in Brauer's height zero conjecture for two primes.

An alternative proof of Brauer's height zero conjecture.

Application of techniques to extend known results.

Abstract

The celebrated It\^o-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup. The principal block case of the recently-proven Brauer's height zero conjecture isolates the abelian part in the It\^o-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture.